Geo_Book_Answers

LESSON 13.3
1. Case 1: P is collinear with A and B. Use the Line
Intersection Postulate to show that P and E are the
same point and use the definitions of bisector and
midpoint to get AP BP.
Case 2: P is not collinear with A
and B. Use the SAS Congruence Postulate to get
AEP BEP. Then use CPCTC to get AP BP.
A
2. Case 1: P is collinear with A and B. Use the
definitions of congruence and midpoint to show
that P is the midpoint of AB.Then use the
definition of perpendicular bisector.
A
C
P
E
D
P
E
Case 2: P is not collinear with A and B.Draw
midpoint E and PE. Use the SSS Congruence
Postulate to get AEP BEP. Then use CPCTC
and the Congruent and Supplementary Theorem
to prove that AEP and BEP are both right
angles. Therefore PE is the perpendicular bisector
of AB by the definitions of midpoint and
perpendicular.
3. Use the reflexive property and
C
the SSS Congruence Postulate to get
ABC BAC. Therefore,
A B by CPCTC.
4. Use the reflexive property and
the ASA Congruence Postulate to
get ABC BAC. Then use
CPCTC and the definition of
isosceles triangle.
5.
B
C A
P
B
B
A B
A B
Draw line BA.
Use the Isosceles Triangle Theorem to get
PAB PBA. Use the Angle Addition
Postulate and the subtraction property to get
BAC ABC. Then use the Converse of the
C
Isosceles Triangle Theorem (CB CA), the reflexive
property, and the SSS Congruence Postulate to get
ACP BCP. Therefore,ACP BCP by
CPCTC.
6.
C m
A
Use the Line Intersection Postulate and the
Perpendicular Bisector Theorem to get AP BP
and BP CP. Then use the transitive property and
the Converse of the Perpendicular Bisector
Theorem to prove that point P is on line n.
7.
C
Use the Line Intersection Postulate and the Angle
Bisector Theorem to prove that Q is equally distant
from AB and AC and from AB and BC. Then use the
transitive property and the Converse of the Angle
Bisector Theorem to prove that point Q is on line n.
8.
B
A
Use the Linear Pair Postulate and the definition of
supplementary angles to get m3 m4 180°.
Then use the Triangle Sum Theorem and the
transitive property to get m1 m2
m3 m3 m4. Therefore, m1
m2 m4 by the subtraction property.
9. D
A
n
m
A B
1
3 4
2
1
2
B
Q
n
P
3 4
C
C
Use the Triangle Sum Theorem and the addition
property to get mA m1 m3 mC
m4 m2 360°. Then use the Angle Addition
Postulate and the substitution property to get
mA mABC mC mCDA 360°.
10. Use the definitions of median
and midpoint to get BM 1 BC 2 and
AN 1
C
N M
AC. 2 Then use the multiplication
property and the substitution
A B
property to get AN BM . By the
reflexive property, the Isosceles Triangle Theorem,
and the SAS Congruence Postulate, ABN
BAM. Therefore, BN AM by CPCTC.
B
ANSWERS TO EXERCISES 147
Answers to Exercises